Prove that an infimum equals a supremum in different sets

Suppose that A is contained in the set of all real numbers and is bounded above. Let U be the set of upper bounds, i.e., U={x is an element of the real numbers: x>=a for all a elements of A}. Prove that U is bounded below and that inf U=sup A. (Note: U is not the empty set by the assumption that A is bounded above.)

Re: Prove that an infimum equals a supremum in different sets

Originally Posted by lovesmath

Suppose that A is contained in the set of all real numbers and is bounded above. Let U be the set of upper bounds, i.e., U={x is an element of the real numbers: x>=a for all a elements of A}. Prove that U is bounded below and that inf U=sup A. (Note: U is not the empty set by the assumption that A is bounded above.)

First we need to say that is not empty, therefore is bounded below.
Let . You need to show that is a lower bound for .
Then show that is the greatest lower bound of .